The oblique contact-impact characteristic of the composite structural bar
composed of Babbitt alloy and low-carbon steel (ZChSbSb11-6 / AISI 1020)
with a solid flat surface (AISI 1045) was studied theoretically and
experimentally. The dynamic equation of the composite structural bar with
vibration response during the contact-impact was established using the
momentum theorem and assumed mode method, and the instantaneous contact
forces during different impact phases were analyzed based on modified
Jackson–Green model. Four sets of experiments (i.e. different proportion of
Babbitt, ξ={1/8,1/2,3/4,7/8}) for the
initial angle, θ=45∘, and different initial velocities
were performed; and, the rebound linear and angular velocity of the contact
point of composite structural bar after impact was calculated and compared
with experimental results. Besides, the coefficient of restitution, the
relation of contact force and contact deflection, and the permanent
deformation were also compared for the composite structural bars with
different proportions in combination, ξ. Three critical angles are
found to determine whether the composite bar slides or not, but are
prominently different for the composite bars with different ξ. In
comparing with the experimental results, the numerical solutions of rebound
linear and angular velocity had yield encourage results and, all relative
errors were small, indicating that the simulations are in good agreement
with the experimental results. Also, the oblique contact-impact behavior
involving the coefficient of restitution, the relation of contact force and
contact deflection, and the permanent deformation was explained in detail.
It can be concluded that as the proportion of Babbitt ξ increases, the
composite structural bar presents a characteristic of ease of deflection.
And the contact-impact behavior of structural entity is closely related to
the inherent properties of the elasto-plastic material, especially for the
weak material of composite structures. The more easily the impacting object
is deformed, the small the contact force during the contact-impact, which
also indicates the yield strength of weak material is a very significant
parameter in the event of collision. Such work could give conducive insights to contact-impact problems of the key parts or structures composed of composite materials in mechanical system.

1Introduction

Contact-impact is a particularly salient phenomenon which plays an important
role pervasive in many applications involving the design and analysis of
mechanical system, such as robotics (Banerjee et al., 2017; Flores-Abad et
al., 2014), iron and steel metallurgy (Y. Wang, et al., 2017a), vehicles
(Yuan et al., 2016), civil structures (H. Wang, et al., 2017a), composite
structures (Mao et al., 2017; Park, 2017), and many other fields (Brake,
2015), since it affects motion parameters of the impacting object (e.g.
rebound linear and angular velocity), contact force, and indentation
deformation involving its patterns etc (Dong et al., 2018; Ghaednia et al.,
2017b). Various normal and oblique impact events happening on the surface of
two or more colliding bodies had been studied by many researches for
decades. For any contact-impact events, authors focus primarily on what
happened during the impacts and/or after the impacts (Meng and Wang, 2018).
On one hand, the behavior of the impacting object such as bar, rod, or
sphere during the impact is hard experimentally to measure, since the impact
occurs in a very shot time and involves large contact force and changes in
velocity of the colliding bodies. On the other hand, to find the relation of
contact force and contact indentation and, simulate and predict the motion
of the impacting object will continue remain relevant challenges in
contact-impact. Additionally, the difference of impact response for
different shapes or material properties is very obvious (Y. Wang et al.,
2017a), especially for composite materials withstanding high impact load,
i.e., contact force (Xie et al., 2016; Zhikharev and Sapozhnikov, 2017).
Numerous studies reported in literature indicated that composite materials
are widely used as structural materials in various fields, such as
aerospace, military, as well as in nuclear power due to their
characteristics of higher strength and stiffness, compared with general
metallic materials (Appleby-Thomas et al., 2015; Boroujerdy and Kiani, 2016;
Li et al., 2016). However, in engineering practice, heavy loads and/or
impact loads with uncertainty are easily to induce varying levels of
internal damage, which can also result in significant degradation of the
structural strength. Thus, more and more investigators and scholars pay
close attention to the contact-impact problems and its effect on the
performance as the key parts or structures are increasingly made of
composite materials.

For the bearing bushing of oil-film bearing as the key load-carrying
component, it is usually a composite structure composed of Babbitt and
steel, and which is most commonly prepared by casting or spraying process
(Babu et al., 2015; J. M. Wang et al., 2017; Zhang et al., 2015).
Especially, the impact damage is a ubiquitous phenomenon during the
operation of oil-film bearing, and composite structures (i.e. Babbitt layer
and steel substrate) with different proportion in combination have
significant differences in their bonding strength (Li et al., 2016; Wang et
al., 2015; Y. Wang, et al., 2017b). So, a systematic study on the
contact-impact behavior of ZChSbSb11-6 Babbitt/20 steel composites is
required to establish a suitable dynamic analysis model in order to ensure
the reliability, safety, and service life of the product.

To the best of our knowledge, although a great deal of research and
publications concerning the fully elastic or elasto-plastic contact-impact
problems have been carried out on the contact/impact models in order to
represent the complicated contact-impact behaviors – such as coefficient
of restitution (Christoforou and Yigit, 2017; Khulief, 2013; Wang et al.,
2019), permanent deformation (Ghaednia et al., 2015b,
2017a; H. Wang, et al., 2017b; Y. Wang, et al., 2017a), and the analytical
relation of contact force-indentation (Bartier et al., 2010; Brake, 2012;
Ghaednia et al., 2016) – of the impacting or colliding objects, those
contact models for impact events can be divided into two major categories:
flattening and indentation models (Ghaednia et al., 2015a). For the
indentation models, such as Kogut–Komvopoulos model (Kogut and Komvopoulos,
2004), Ye–Komvopoulos model (Ye and Komvopoulos, 2003), and Brake model
(Brake, 2012; Brake, 2015), the impacting object is considered to be rigid
and the flat surface deforms. On the contrary, for the flattening model,
such as Vu–Quoc model (Vu-Quoc et al., 2000), Kugut–Etsion model (Kogut and
Etsion, 2002), and Jackson–Green model (Jackson and Green, 2006), the flat
surface is assumed to be rigid and the impacting object deforms. Besides,
there are a few models such as Gheadnia model (Gheadnia et al., 2015a, 2017b) and Ma–Liu model (Ma and Liu, 2015), in which they
assume both objects deform; and, there are many visco-elasto-plastic models
such as Hunt–Crossley model (Hunt and Crossley, 1975) and Christoforou–Yigit
model (Christoforou and Yigit, 2017), that are widely used to study
elasto-plastic impact issues using damping. On the whole, these previously
mentioned models divide the contact-impact events into two main phases: the
loading and unloading phase, where only different description of sub-phases
are used for the loading phase depending on different contact models (Meng
and Wang, 2019). Some models (e.g. Jackson–Green and Kogut–Etsion models)
divide the loading phase into two sub-phases: the fully elastic and the
elasto-plastic phases; and, the loading phase of another model (e.g. Brake
model) consists of three sub-phases: the fully elastic, the elasto-plastic,
and the fully plastic phase. Obviously, the fully plastic regime is the one
limit of the elastic-plastic phase. For the unloading phase, it refers to
the restitution phase. Inspired by these analytical contact-impact models,
more efforts on the application of composite structures will still need to
be taken on the oblique contact-impact behavior, where different proportions
of each material constituting the composite structure should be considered.

In our previous work, we studied the mechanical properties of the
ZChSbSb11-6 Babbitt/20 steel composites with different proportions of
Babbitt, and found that the elastic modulus of composite varies with
different proportions of Babbitt based on the experimental results (Li et
al., 2016). In Y. Wang, et al. (2017a), the relationships of
coefficient of restitution, permanent deformation, and initial impact
velocity during normal impacts were analyzed; and, the constitutive
behavior between the contact force and the contact deflection (i.e.
indentation deformation) was also studied using the established indentation
contact-impact model based on the empirical formulation developed by Brake (2015). Similarly, each contact-impact event is divided into three
phases: the elastic, the elasto-plastic, and the restitution phase. In the
present work, we focus on modeling and simulating the oblique elasto-plastic
impact for composite bars using the modified Jackson–Green model (Ghaednia
et al., 2015a, 2017b), and performing the experiments from
different heights (or initial impact velocities) for the initial impact
angle θ=45∘. Contact-impact behaviors – rebound
velocity, coefficient of restitution, analytical relation of contact force
and contact deflection, and permanent deformation – of the ZChSbSb11-6
Babbitt/20 steel composites with different proportions of Babbitt are also
presented, and the whole work is driven by the goal to get a more complete
understanding of the mechanics property for the composite structures during
contact-impact events.

The layout of this work is that, in Sect. 2, the mathematical models are
established for the oblique contact-impact of composite structural bars with
a solid flat surface, consisting of the dynamic model with vibration
response and the elastic-plastic model of contact force during each impact
phase; in Sect. 3, the specimens used, the experimental methodology, and
the experimental data processing are presented; in Sect. 4, some
comparison analysis between the simulations and the experimental results are
performed; some conclusions are drawn which are useful for the performance
evaluation of composite structures in elasto-plastic oblique contact-impact
events.

2Dynamics of the Oblique Contact-impact

As illustrated in Fig. 1, a composite structural bar B with a length L
impacts a solid flat surface S. The composite bar is made of Babbitt layer
(Tin-based Babbitt, ZChSbSb11-6) and steel substrate (20 steel) as shown in
Fig. 1a, and the solid flat surface is made of 45 steel. A global
reference frame RF(0) of Cartesian unit vectors [i0,
j0, k0] and a mobile reference frame RF(1) of
Cartesian unit vectors [i1, j1,
k1] are considered as shown in Fig. 1b.

The initial impact angle is θ. The center of the mass of the
composite bar is point C, the contact point is at E, and the top point is A.
The gravitational force is G and the contact force during the
impact is F. Generalized coordinates q1 , q2 , q3 are
employed to characterize the instantaneous configuration of top point A of
the composite bar B in RF(0). The first generalized coordinate q1 denotes
the distance from A to the vertical axis of RF(0) and the second generalized
coordinate q2 denotes the distance from A to the horizontal axis of
RF(0). The third generalized coordinate q3 designates the initial impact
angle θ. To characterize the motion of B in RF(0), the corresponding
generalized speeds defined as ur=q˙r (where r=1,2,3),
represent the rate of change of the generalized coordinates with respect to
time t.

Figure 1

Schematic of the composite structural bar during a
contact-impact event: (a) the 3-D view, and (b) the kinematic chain.

Figure 2

Variation curve of elastic modulus with proportion of
Babbitt.

2.1Elastic modulus of the composite bar

For the elastic modulus Eb of composite bars with different proportion
of Babbitt ξ, the experimental results in Li et al. (2016) has
been used as shown in Fig. 2. In Fig. 2, the elastic modulus (Eb) of
composite bars decreases monotonically with the increasing proportion of
Babbitt (ξ), and the pentagram points represent the elastic modulus of
specimens used in our experiments (as seen in Sect. 3.1).

The mathematical expression for the elastic modulus (Eb) depending on
proportion of Babbitt (ξ) provided in Li et al. (2016) is as
below:
1Eb=375.93ξ3-493.61ξ2-45.53ξ+213.50.
with
2ξ=ABabbittAb
where ξ represents the proportion of Babbitt, Ab is the
cross-section area (as shown in Fig. 1a) of the composite bar, and
ABabbitt is the area of Babbitt.

2.2Dynamic model with vibration response

Published amounts of research indicated that the effect of longitudinal
and/or transverse vibration response during the motion and dynamic modeling
can not be neglected for the elasto-plastic contact-impact events
(Bazrafshan et al., 2014; Shafei et al., 2018). Authors previous works had
done a quantitative analysis for mechanical vibration of the impacting
object during impact events, and the longitudinal and transverse vibration
responses were found to effect the normal and tangential velocities of the
contact point after the impact, respectively. Besides, for the oblique
contact-impact with sliding, considering the response of longitudinal and
transverse vibration at the same time was more reasonable than other cases,
such as only longitudinal or transverse vibration. But, increasing the
number of shape functions did not effect the result significantly (Meng and
Wang, 2018; Wang et al., 2019). Thus, vibration response with two shape
functions is considered in the deformation of the contact point. In what
follows, the equations of motion of a composite structural bar with a solid
flat surface during the oblique contact-impact events were developed using
the momentum theorem and assumed mode method.

The unit vectors of the reference frames can be expressed as:
3i0j0k0=i1j1k1R10,
where R10 represents the transformation matrix
R10=c3-s30s3c30001
and s3=sin⁡q3, c3=cos⁡q3.

The angular velocity and acceleration of the composite bar B are:
4ω=00u3,α=00u˙3.
The position and the velocity vector of the point A are:
5rA=q1q20,vA=u1u20.
As previously mentioned above, longitudinal and/or transverse vibration of
the composite bar B during the contact-impact events will be considered in
the deformation of contact point. Using the assumed mode method, the
longitudinal elastic displacement ν of an arbitrary point P on the composite bar in RF(1) can be expressed as:
6ν(x,t)=∑i=1nΦi(x)qi(t),
with the longitudinal shape modes are:
7Φi(x)=cos⁡(i-1)πLx.
where n represents the number of vibrational modes selected, Φi(x) is the mode shape of a composite bar with both ends free
and qi(t) is the elastic generalized coordinate.

Similarly, the transverse elastic displacement y of an arbitrary point P on the composite bar in RF(1) can be expressed as:
8y(x,t)=∑i=1nΦi′(x)qi(t),
with the shape modes for transverse vibration are:
9Φi′(x)=cosh⁡λiLx+cos⁡λiLx-β(λi)sinh⁡λiLx+sin⁡λiLx,
and
10β(λi)=cosh⁡λi-cos⁡λisinh⁡λi-sin⁡λi.
where Φ′i(x) represents the mode shape of a composite bar with
both ends free, and λi is the consecutive root of the
characteristic equation.

So, for any point P of the composite bar, the vector rAP can be expressed in terms of the mobile reference frame RF(1) as:
11rAP(1)=x+ν(x,t)i1+y(x,t)j1,
In the global reference frame (0), the position vector rAP can
be written as:
12rAP(0)=rAP(1)⋅R10.
The velocity of any point P of the composite bar B has the following
expression in the reference frame (0):
13vP=vA+ddtrAP(0),
The kinetic energy of the composite bar, T is calculated as follows:
14T=ρ2∫0LvP⋅(vP)Tdx,
where ρ is mass per unit length.

Assume that the duration of the system involved in an contact-impact event
is [t1, t2]. Using the momentum theorem, the generalized momentum
and impulse equations for the impact of a composite bar B with a solid flat
surface S are found after the integration of Kane's equations (Kane and
Levinson, 1985) which yields,
15Pj≈Mj(t2)-Mj(t1),
where Pj is the generalized impulse, Mj is the generalized moment,
and j represents the jth generalized speeds.

The kinetic energy in (0) is a function of q1,…,qn+3,
u1,…,uj and t. The generalized moment Mj can be calculated as:
16Mj(t)=∂T∂uj,j=1,…,k
where k is the number of generalized speeds.
The generalized impulse Pj is calculated as:
17Pj=∫t1t2∂vE∂uj⋅Fdt,
where ∂vE/∂uj is the partial velocity at any time.

Figure 3

Graphic of the velocity vector for two collision bodies
during an impact event.

Substituting Eqs. (16) and (17) into Eq. (15) yields the equation for
the oblique contact-impact:
18∂vE∂uj∫t1t2Fdt=∂T∂uj(t2)-∂T∂uj(t1).
To describe the motion of the system at t2, more information must be
added to Eq. (18). Figure 3 shows the graphic of the velocity vector
for two bodies during the contact-impact events. The bodies D1 and
D2 impact each other and their contact points are E1 and E2,
respectively.
The velocity of approach va is:
19va=vE1(t1)-vE2(t1),
where vE1(t1) and vE2(t1) are the velocities of
points E1 and E2 at time t1, respectively.
The velocity of separation vs is:
20vs=vE1(t2)-vE2(t2),
where vE1(t2) and vE2(t2) are the velocities of
points E1 and E2 at time t2, respectively.

As shown in Fig. 3, the contact force exerted on D1 by D2 at the
contact point E during the contact-impact, F, is integrated with respect to t from t1 to t2 and resolved into two mutually perpendicular
components, normal impulse, Fn and tangential impulse, Ft. The
normal components of va and vs have opposite
directions, and the relation between their magnitudes is based on the
definition of the coefficient of restitution e:
21j0⋅vs=-ej0⋅va.
A condition, Δ, is used to find the type of the friction for the
contact-impact of the composite structural bar B on a solid flat surface S, which can be estimated by the formula as below:
22Δ=Ft-μsFn,
where μs is the coefficient of static friction between a composite bar and a flat surface.

When Δ<0, the tangential component of vs must be zero
which means:
23j0×vs×j0=0.
Thus, no sliding occurs for this condition. Adding Eqs. (21) and (23)
into Eq. (18) yields the motion of the composite bar during the
contact-impact with all unknowns.

When Δ>0, the composite bar slides during the impact,
and the following relation must be satisfied:
24Ft=-μkFnj0×vs×j0j0×vs×j0.
where μk is the coefficient of kinetic friction between a
composite bar and a flat surface.

In this case, the kinetic of the composite bar with all unknowns can be
calculated by adding Eqs. (21) and (24) into Eq. (18). MATLAB has been used
to find and solve the above equations of motion. Because the impulse is the
cumulative effect of contact force on time and is a process quantity, the
instantaneous contact force during the contact-impact can not be obtained,
and further studies are needed to characterize the relation of contact force
and contact deflection.

2.3Contact force analysis

Based on flattening model, i.e., modified Jackson–Green model provided in
Gheadnia et al. (2015a, 2017b), the contact-impact
process of the composite bar with a solid flat surface is divided into three
phases: the elastic, the elasto-plastic, and the restitution phase. In what
follows, the instantaneous contact forces during different impact phases are
analyzed.

Figure 4

Schematic of the contact deflection during impact events.

Figure 4 shows the schematic of contact deflection during impact events. The
contact-impact process can be described as follows. When the rounded end of
one impacting object (i.e. the composite structural bar) is brought into
contact with the solid flat surface, the elastic phase starts and continues
until yielding of weak material occurs; then, the elasto-plastic phase
starts and continues until the contact point E of the impacting object stops
(i.e. the normal velocity of the contact point is zero at this moment). That
is to say, the indentation of the contact point E into the solid flat surface
reaches the maximum deformation. At this point, the elasto-plastic phase
comes to an end and the restitution phase starts. The restitution phase
continues until the contact force reaches zero (i.e. there is no contact at
this point), and at this instance the deformed region reaches the permanent
deformation.

Using the methodology originated from Hertzian theory of the contact, the
contact force for the elastic phase of the contact, Fe, is calculated as
follows:
25Fe=43E′R0.5δ1.5,
with
261E′=1-μ12Eb+1-μ22Ef,
and
27δ=yE±ν(L,t)⋅sin⁡θ±y(L,t)⋅cos⁡θ.
where R is the reduced radius (Note that R=Rb for our case since
Rf=∞), E′ is the reduced modulus of elasticity,
the contact deflection is δ, yE is the y coordinate of the
contact point E in (0), and Eb, μ1 and Ef, μ2 are
the elastic modulus and Poisson's ratio of the two materials in contact (B
and S) respectively.

According to Gheadnia et al. (2015a, 2017b),
the effective elasto-plastic phase starts at δ*≥1.9,
which also means the elastic phase will end when the critical point of the
elastic phase is δ*=1.9, and will provide the initial
conditions for the next phase, where δ*=δR2E′πCjSy2, and Cj=1.295e0.736μ.

For the elasto-plastic phase, the expression of contact force, Fp, is shown as below:
28Fp=Fce-0.17⋅δ*5/12⋅δ*1.5+4HGCjSy1-e-178δ*5/9⋅δ*1.1,
where Fc is the critical force at the instant the yield occurs,
HG is the average normal pressure, the real contact radius is a, and
δc is the critical deformation,
with
Fc=43RE′2πCjSy23,HGSy=2.84-0.921-cos⁡πaR,B=0.14e23⋅Sy/E′,anda=R2πCjSy2E′2δ1.9δcB.

For the restitution phase, it has also been assumed to follow the Hertzian
theory. So, the contact force for restitution phase, Fr, is calculated as follows:
29Fr=43E′Rr0.5δ-δr1.5,
where Rr is the new radius of curvature, and δr is the permanent deformation. Then,
30δrδm=0.81-δm/δy+5.56.5-2,
where δm is the maximum deformation when the normal velocity of the contact point E of the composite bar is zero, and δy is
indentation at which yield starts. From continuity, the new radius of the
curvature Rr in this phase is changing as below:
31Rr=1δm-δr33Fm4E′2.
where Fm is the maximum contact force corresponding to the maximum
deformation δm.

3Experimental methodology

The oblique contact-impact events were performed by dropping the composite
structural bar (B) from different initial heights (i.e. initial impact
velocities) with an initial impact angle θ on a solid flat surface
(S). A 3-D high-speed camera was adopted to capture the kinematic data of the markers on the composite structural bar during contact-impact events. Four sets of experiments (i.e. ξ={1/8,1/2,3/4,7/8}) for the initial angle, θ=45∘, and
different initial velocities have been completed. Each height was tested at
least three times in order to achieve stable experimental data.

Figure 5

Shape and dimension of specimens for contact-impact
testing, with different values of h described in Table 1.

A rounded ended composite bar B was composed of ZChSbSb11-6 Babbitt and 20
steel (i.e. Babbitt layer and steel substrate) with length L, diameter d,
elastic modulus Eb, and yield strength of σyb. The solid
flat surface S is made of 45 steel with length Lf, width Wf, thickness Tf, and with elastic modulus Ef and yield strength of
σyf. The tolerance for the diameter of one of the rounded ended
bar is ±0.02 mm. The surface roughness Ra of the solid flat
surface is 1.6 µm. Four sets of the composite bar with different
proportion of Babbitt ξ were prepared and used to do the impact
experiments. Figure 5 shows the shape and dimension of specimens for
contact-impact testing, with different proportion of Babbitt ξ. The
specification of specimens are described in Table 1. Using CNC machine
tools, specimens for processing were completed, of which the quality
coincided well with the requirements; then, these specimens were cleaned
thoroughly with alcohol and dried in air. The microscopic characterization
of specimens for contact-impact testing is shown in Fig. 6.

Figure 6

Microscopic characterization of specimens.

4.1Impact testing

The specimens should be pretreated before the test. The procedure of setting
markers on the composite bar was as below. First, the body of composite bar
was painted into black except for both ends, and both ends were covered
using insulating tapes. Then, after completing the painting process, the
insulating tapes on both ends were removed, and seven markers (white color)
were arranged in turn on the body of each composite bar. The gap between the
markers were distributed as uniform as possible, as shown in Fig. 7.

The markers were used to track the displacement and the angle of the
composite bar during contact-impact events. All black environment was used
to support the markers' detection so that it was easy to identify the
markers in the video images. Later, the captured video images were analyzed
using digital image processing technology. In this study, four sets of
experiments (i.e. ξ={1/8,1/2,3/4,7/8}) performed from different initial drop heights for the initial angle,
θ=45∘, have been completed. The range of drop heights
is 0.05–0.6 m. In order to ensure the accuracy of experimental results and
reduce the influence of systematic error, impacts should be independent of
each other. Each captured video image contained about 1500 frames, and each
frame contained 512×252 pixels. In what follows, taking one of the
experiments with θ=45∘ (which the drop height H=0.10 m,
and the composite bar with proportion of Babbitt ξ=0.5, with length
L=80 mm, and diameter d=10 mm) as an example, the data were analyzed.

4.2Experimental data processing

The kinematic data such as the impact angle of the composite bar at the
instant of impact, and the before- and after-impact velocities, can be
obtained using MATLAB. The MATLAB function, [B, L] = bwboundaries(BW), is
first of all used to find the boundaries of each marker, and the central
point of each marker can be solved by averaging all boundaries, as shown in
Fig. 4.

Figure 8

Motion analysis of the composite bar analyzed with
digital image processing method. Note that red boundaries and diamond
symbols represent the profile and the central point of each marker,
respectively.

Here, the motion data of the composite bar during the contact-impact event
has been explained in detail. The position of the closest center point
rc(1), the velocity of the closest center point vc(1), and the
angular velocity of the composite bar ω are determined and shown in
Figs. 9–11 respectively.

Figure 9

Position of the central point of marker 1 before and
after the impact.

Figure 10

Velocity of the central point of marker 1 before and
after the impact.

Figure 10 shows the tangential and normal components of the position and
velocity of the closest center point. For this case, the impact occurs at
t≈0.1631 s. The normal velocities before and after the contact-impact are -1.61353 and -0.472 m s-1 respectively. The tangential velocities before and after the contact-impact are 0.01282 and -0.0639 m s-1
respectively. Moreover, the slope of normal velocities before the
contact-impact is 9.853 m s-2, which matches the standard gravitational
acceleration g (g=9.80665 m s-2), so it also indirectly reveals that the experimental data are reliable.

Figure 11

Angular velocity of the composite bar before and after
the impact.

As shown in Fig. 11, the angular velocity before and after the
contact-impact are ωb=0.00581 rad s-1 and ωa=-21.878 rad s-1 respectively. The angular velocity presents hardly any differences before the impact occurs, but there are slight fluctuations.

5Results and discussion

In this work, the numerical simulations of the dynamic model of the
composite structural bar with a solid flat surface during the oblique
contact-impact events have been compared with the experimental results in
terms of the linear and the angular motion, i.e., the normal and tangential
velocities of the contact point after the impact, and the rebound angular
velocities of the impacting object (i.e. composite structural bar). In
addition, the coefficient of restitution, the relation of contact force and
contact deflection, and the permanent deformation were also compared for the
composite structural bars with different proportions of Babbitt. Table 2
shows the material properties and the geometrical dimensions used for the
numerical simulations.

Table 2

The material properties of the composite structural bar and
the solid flat surface.

Figure 12 depicts Δ, which shows the condition to determine whether
the composite structural bar slides or not on the solid flat surface, as a
function of initial impact angle. All curves with different proportion of
Babbitt present the approximate W distribution in the interval (0,
90∘), and three critical initial impact angles (at Δ=0)
are found. For the composite structural bar with ξ=1/8 (i.e. the
legend of black square), three critical initial impact angles are
27.3, 37.4, and 66.3∘ respectively. From
the variation curves shown in Fig. 12, we can also know that near the first
lowest point, as the proportion of Babbitt, ξ, increases the first and
second critical initial impact angle, θ, decreases. On the contrary,
as the proportion of Babbitt, ξ, increases the third critical initial
impact angle, θ, increases. That is to say, the critical angles are
prominently different for the composite structural bar with different ξ
during contact-impact events.

Figure 12

The condition for sliding or no sliding as a function of
initial impact angle, with the properties described in Table 2, and with
ξ varied from 1/8 to 7/8.

5.2Rebound velocity

The tangential and normal velocities of the contact point after the impact
with θ=45∘ for different proportions of Babbitt are
compared with the experimental results, and the results are shown in Figs. 13–14. For each specimen with different proportion of Babbitt, the maximum
and average relative errors of the rebound normal and tangential velocity
are also calculated by comparing the numerical value with the experimental
results. All simulation results almost depict the same trend with the
experimental data.

Figure 13

Comparison between the simulations and the experimental
results for the rebound normal velocity: (a)ξ=1/8, (b)ξ=1/2,
(c)ξ=3/4, and (d)ξ=7/8.

Figure 13 presents the rebound normal velocity increases as the initial
impact velocity increases. All of the specimens show same trend, while
numerical values are slightly different from the experimental results. For
ξ=1/8 shown in Fig. 13a, the maximum and average relative errors of
the rebound normal velocity are 4.69 % and 2.14 % respectively. For
ξ=1/2, 3/4, and 7/8 shown in Fig. 13b–d, the maximum and
average relative errors of the rebound normal velocity in turn are 4.49 %
and 2.98 %, 4.16 % and 2.12 %, 5.05 % and 2.56 %, respectively.

Figure 14

Comparison between the simulations and the experimental
results for the rebound tangential velocity: (a)ξ=1/8, (b)ξ=1/2, (c)ξ=3/4, and (d)ξ=7/8.

Figure 14 shows the rebound tangential velocity of the composite bar for
different initial impact velocities and with the different proportion of
Babbitt. Similar to the rebound normal velocity results, the rebound
tangential velocity increases as the initial impact velocity increases with
the same trend for each specimen. However, all curves exist obvious
fluctuations, which indirectly indicates that the effect of friction and
what role of it in contact-impact events are needed to be further explored.
For ξ=1/8, 1/2, 3/4, and 7/8 shown in Fig. 14a–d, the maximum
and average relative errors of the rebound tangential velocity in turn are
4.84 % and 2.58 %, 4.92 % and 3.21 %, 4.97 % and 3.22 %,
4.96 % and 2.72 %, respectively.

Figure 15

Comparison between the simulations and the experimental
results for the rebound angular velocity: (a)ξ=1/8, (b)ξ=1/2, (c)ξ=3/4, and (d)ξ=7/8.

Furthermore, the angular velocity of the composite structural bar after
impact has been analyzed and compared with the experiments, as shown in
Fig. 15. The rebound angular velocity also increases with the increase of
initial impact velocity for each specimen. No significant difference is seen
between the simulations and the experiments. All the average relative errors
are less than 5 %, and the maximum error for different proportion of
Babbitt does not exceed 15 %, indicating that the established model is
effective. From the above analysis, it can be seen that all relative errors
of the established dynamic model are small, which reveals that the
simulations are in good agreement with the experimental results.

5.3Coefficient of restitution

As is well known, the coefficient of restitution is a key parameter, which
reflects the displacement restoring capacity of colliding bodies during the
contact-impact. Based on its definition, the coefficient of restitution,
e, can be calculated and averaged for all of the experiments using the
following equation:
32e=vEfvEb.
where vEb and vEf are the normal velocity of the contact point E of the composite bar before and after the contact-impact, respectively. The
velocity of the solid flat surface before and after the contact-impact is
considered equal to zero.

Figure 16

The coefficient of restitution as a function of initial
impact velocity, and with ξ varied from 1/8 to 7/8.

Figure 16 shows the coefficient of restitution, e, for different initial
impact velocities, v, from experiments with different proportion of Babbitt, ξ. Experimental results indicate that all curves with ξ varied from 1/8 to 7/8 show the same trend, and the coefficient of restitution decreases as the initial velocities increases. In addition, the coefficient of restitution e, decreases with the increase of the proportion of Babbitt, ξ. This also reveals that as the proportion of Babbitt, ξ,
increases the composite structural bar presents a characteristic of ease of
deflection. Obviously, the behavior of structural entity is mostly governed
by the weak or soft material of composite structures.

5.4Contact force

Figure 17 shows the relationship of contact force and contact deflection for
each specimen with different proportion of Babbitt in the case of maximum
drop height H=0.60 m. In comparing with each specimen, the variation curves
show similar trends in the whole contact-impact. For all of the curves the
contact force starts at zero and increases with an increasing rate until the
maximum deformation (δm) happens; and then decreases with a
decreasing rate until the end of the contact-impact, i.e., at this instance
the contact deflection (δ) of the contact point E reaches the
permanent deformation (δr). Take the curve of ξ=1/8 as
an example, the relative error for the maximum deformation (δm=66.05µm) and the permanent deformation (δr=61.12µm) is 7.46 %; and, it is also the maximum relative error within 10 % for all curves shown in Fig. 17. In addition, as the proportion of Babbitt (ξ) increases the maximum contact force
(Fm) decreases, but the permanent deformation (δr)
increases. This indicates that the contact-impact behavior of structural
entity is closely related to the inherent properties of the elasto-plastic
material, especially for the weak material of composite structures.

Figure 17

The contact force as a function of contact deflection
for H=20 cm, and with ξ varied from 1/8 to 7/8.

Figure 18

The permanent deformation as a function of initial
impact velocity, and with ξ varied from 1/8 to 7/8.

Figure 18 shows the permanent deformation, δr, for different
initial impact velocities, v, from experiments with different proportion of
Babbitt, ξ. As the initial impact velocity (v) increases, the permanent
deformation (δr) increases; and while with the increase of
proportion of Babbitt (ξ), the permanent deformation (δr)
also increases. For the maximum contact force during contact-impact events,
as the initial impact velocity (v) increases, Fm increases; however, as
proportion of Babbitt (ξ) increases, Fm decreases. Thus it can be
seen that the more easily the impacting object is deformed, the small the
contact force during the contact-impact, which also indicates the yield
strength of weak material is a very significant parameter in the event of
collision.

Experimental studies have shown that the solid flat surface has no deformed
region and only the contact deflection happens on the composite bar, which
is well matching the flattening models. These phenomena can indirectly
illustrate the accuracy of established model based on the flattening model.
For the indentation models, we can use a profilometer or a CLSM (confocal
laser scanning microscope) to measure the profile of the deformed region
(i.e. permanent deformation) on the solid flat surface after each impact.
But so far, there is still no effective way to measure the contact
deflection on the composite structural bar. So, additional experimental
studies of permanent deformation during contact-impact are needed to perform
in order to get a wider range of established model such as to predict the
contact force and contact deflection.

6Conclusions

In this study, the oblique contact-impact of the composite structural bar
composed of Babbitt layer and steel substrate with a solid flat surface has
been analyzed theoretically and numerically. The dynamic motion of the
composite structural bar with vibration response during the contact-impact
has been established using the momentum theorem and assumed mode method. The
instantaneous contact forces during different impact phases were also
analyzed based on modified Jackson–Green model. Four sets of experiments
(i.e. different proportion of Babbitt, ξ={1/8,1/2,3/4,7/8}) for the initial angle, θ=45∘,
and different initial velocities have been completed. Then, the rebound
linear and angular velocity of the contact point of composite bar after
contact-impact has been calculated and compared with experimental results.
In addition, the coefficient of restitution, the relation of contact force
and contact deflection, and the permanent deformation were also compared for
the composite structural bars with different ξ.

It has been shown that three critical angles are found to determine whether
the composite bar slides or not, but are prominently different for the
composite structural bar with different ξ. The comparison for rebound
linear and angular velocity between the simulations and the experimental
results had yield encourage results, which revealed that the simulations are
in good agreement with the experimental results. Moreover, the oblique
contact-impact behavior involving the coefficient of restitution, the
relation of contact force and contact deflection, and the permanent
deformation was explained in detail. First, for the coefficient of
restitution, as the proportion of Babbitt, ξ, increases the composite
structural bar presents a characteristic of ease of deflection. Obviously,
the behavior of structural entity is mostly governed by the weak or soft
material of composite structures. Then, for the relation of contact force
and contact deflection, the contact-impact behavior of structural entity is
closely related to the inherent properties of the elasto-plastic material,
especially for the weak material of composite structures. Lastly, for the
permanent deformation, the more easily the impacting object is deformed, the
small the contact force during the impact, which also indicates the yield
strength of weak material is a very significant parameter in the event of
collision. Further, due to the limit of measuring the contact deflection on
the composite bar, more studies are needed to perform in order to get a
wider range of established model.

Data availability

The data can be made available upon request.
Please contact Yao Wang (sjtuyao@sjtu.edu.cn) or Zhuang Fu
(zhfu@sjtu.edu.cn).

YW conceived the overall idea of this paper,
and performed the specimen fabrication and experiments, and analyzed the
data. ZF verified the established model and supervised the whole work. YW
and ZF wrote the paper and agreed on the final form of the manuscript.

Competing interests

The authors declare that they have no conflict of interest.

Acknowledgements

I would like to thank my PhD supervisor Wenjun Meng for his support and feedback provided on this work and gratefully
acknowledge to the Oil-film Bearing Branch of Taiyuan Heavy Machinery Group
Co., Ltd,. In particular, I also want to gratefully acknowledge Dan B. Marghitu and his research team at Auburn University for the helpful guide on the field of contact mechanics.

Financial support

This research has been supported by the National Natural Science Foundation of China (grant no. 51875333) and the Project of STCSM (grant no. 17441901000).

Review statement

This paper was edited by Guimin Chen and reviewed by two anonymous referees.